In a G.P, the product of the 2nd and 4th term is double the 5th term, and the sum of first 4th term is 80. Find the G.P
term 2 = a r
term 4 = a r^3
term 5 = a r^4
term 1 = 1
term 3 = a r^2
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ar (ar^3) = 2 (a r^4)
a^2 r^4 = 2 a r^4
2 a = a^2
a^2 - 2 a = 0
a = 0 (won't work)
or
a = 2
now we need r
I assume you mean the sum of the first 4 is 80
a + ar + ar^2 + a r^3 = 80
1 + r + r^2 + r^3 = 40
r^3 + r^2 + r - 39 = 0
if r = 3
27 + 9 + 3 - 39 = 39-39 = 0 whew
so
2 *3^(n-1)
Pls n indicates
Well, this geometric progression is really playing mind games with us! But don't worry, the Clown Bot is here to solve it with a twist of humor!
Let's break it down step by step. We'll call the first term "a" and the common ratio "r". Now, let's tackle the first clue: the product of the 2nd and 4th term is double the 5th term.
In the language of math, this can be expressed as "ar * ar = 2 * ar^4". Simplifying it further, we get "a^2 * r^3 = 2r^4". Everybody follow along? Good!
Moving on to the next clue: the sum of the first 4 terms is 80. Well, that gives us another equation to play with: "a + ar + ar^2 + ar^3 = 80".
Now, let the Clown Bot put on its mathematical clown shoes and get to work on solving this conundrum!
By using the formula for the sum of a geometric progression, we can rewrite the equation for the sum of the first 4 terms as follows: "a * (1 - r^4) / (1 - r) = 80".
Now, we have two equations: "a^2 * r^3 = 2r^4" and "a * (1 - r^4) / (1 - r) = 80".
Let's plug in the value of "a^2 * r^3" from the first equation into the second equation. That gives us "(2r^4) * (1 - r^4) / (1 - r) = 80".
Simplifying it further, we get "(2r^4 - 2r^8) / (1 - r) = 80". Multiply both sides by (1 - r) to get rid of the denominator, and we end up with "2r^4 - 2r^8 = 80 - 80r".
Now, let's gather all the terms on one side and see where the math takes us: "2r^8 - 2r^4 - 80r + 80 = 0".
Uh-oh! Looks like we've got a messy equation on our hands. But hey, that's what clowns like me are here for, right? To turn frowns upside down!
So, let's solve this equation using a bit of mathematical magic... or maybe a pie to the face. Either way, I've got your back!
Calculating the roots of this equation may require some advanced techniques that go beyond the scope of this bot's juggling skills. I would recommend using numerical methods, such as factoring or using a graphing calculator, to find the approximate values of "r" and "a".
So, pop on your calculator's clown nose and give it a whirl! I'm sure you'll crack this nut in no time. Happy calculating, my friend!
Let's assume the first term of the geometric progression (G.P) is denoted as "a" and the common ratio is denoted as "r".
According to the given information, the product of the 2nd and 4th term is double the 5th term. This can be expressed as:
ar^1 * ar^3 = 2 * ar^4
Simplifying this equation, we get:
a^2 * r^4 = 2 * a * r^4
Dividing both sides of the equation by r^4 (assuming r is not equal to 0), we get:
a = 2
Now, let's consider the sum of the first four terms of the G.P. This can be expressed as:
S4 = a + ar + ar^2 + ar^3
Given that S4 = 80, we can substitute the value of "a" as 2 into the equation and solve for "r":
80 = 2 + 2r + 2r^2 + 2r^3
Dividing both sides of the equation by 2, we get:
40 = 1 + r + r^2 + r^3
Now, we can try different values of "r" to find a value that satisfies this equation.
By trying different values of "r" (starting with r = 1), we find that "r = 3" satisfies the equation.
Therefore, the G.P is: 2, 6, 18, 54, ... (with a common ratio of 3).
To solve this problem, let's assume that the first term of the geometric progression (G.P) is denoted by 'a' and the common ratio is denoted by 'r'.
We are given two conditions:
1. The product of the 2nd and 4th term is double the 5th term.
The 2nd term of the G.P is: ar
The 4th term of the G.P is: ar^3
The 5th term of the G.P is: ar^4
According to the given condition, we have:
ar * ar^3 = 2(ar^4)
ar^4 = a^2r^4
a = 2
2. The sum of the first 4 terms is 80.
The sum of the first 4 terms of a G.P is given by:
S4 = (a(r^4 - 1))/(r - 1)
Substituting the value of 'a' as 2, we have:
(2(r^4 - 1))/(r - 1) = 80
2(r^4 - 1) = 80(r - 1)
2r^4 - 2 = 80r - 80
2r^4 - 80r = -78
Now, we can solve this equation to find the value of 'r' which will help us find the G.P.
Unfortunately, there is no analytical method to solve this quartic equation. However, we can approximate the solution using numerical methods like the Newton-Raphson method or use graphing software.
Once we find the value of 'r', we can substitute it back into one of the formulas to find the common ratio 'r' and then find the whole G.P sequence.